Although it's quite closely related to German, Dutch gives the impression of being one of the odder languages in Europe due to its distinctively peculiar system of vowel sounds. Most people know little or nothing about the Dutch language except that it's similar to German, thus there's is a common tendency to use German pronunciation rules for Dutch names. This approach often produces results that are phonetically wrong but generally acceptable (e.g. Frans Brüggen, see below). However, it cannot always be applied: Dutch contains many odd looking vowel combinations that German lacks, and it also pronounces certain vowels very differently. When German rules can be applied, you should not assume they are correct.

Flemish, the variety of Dutch spoken in Belgium, is the same language for radio purposes. The only notable difference is in the pronunciation of the letter w; the Flemish way matches English, while the Dutch way matches German (see below).

Dutch uses the umlaut on the letter y, but it has a very different meaning and origin from
the German umlaut: Dutch ÿ is originally a contraction of ij. Of course, the
umlaut is not always printed, which tends to cause some uncertainty as to how one should pronounce y.

Like German, some Dutch vowels sound differently depending on whether they are in "open" or "closed" syllables. A closed syllable is one that ends with a consonant, i.e. the English word "fountain" consists of two closed syllables, while the word "futon" has an open syllable followed by a closed syllable. In Dutch the difference between open and closed affects the sounds of a and e, though radio announcers can get away with ignoring the difference.

Do keep in mind that the Dutch u is like the French u: in Germanic terms, that means we pronounce it as though there's an umlaut.

These are not technically exceptions, they are actually names that have become so well known in their incorrect (usually Germanicized) pronunciations that you are better off pronouncing them incorrectly. Thus the following indications do not follow the rules stated above:

Radio Netherlands has created a website in the same spirit as this one, containing a list
of Dutch names with RealAudio recordings of these names being pronounced by a Dutch
radio announcer. These samples are educational to listen to, but there are two things to beware of:

Only names of composers and artists who record with the Radio Netherlands record label are
included, and most of them are fairly obscure.

The audio files demonstrate EXACT Dutch pronunciation, which is not appropriate for
English language radio (see the antiperfection manifesto).